Faraday Symp. Chem. SOC.,1983 18 103-114 Viscoelastic Studies of Reptational Motion of Linear Polydime thylsiloxanes J. LAMB,G. HARRISON BY R. R. RAHALKAR,~ AND A. J. BARLOW Department of Electronics and Electrical Engineering The University Glasgow G12 SQQ Scotland AND W. HAWTHORNE AND J. A. SEMLYEN Department of Chemistry University of York Heslington York YO1 5DD AND A. M. NORTHAND R. A. PETHRICK* Department of Pure and Applied Chemistry University of Strathclyde Thomas Graham Building 295 Cathedral Street Glasgow G1 1XL Scotland Received 23rd August 1983 Viscoelastic measurements are reported on linear polydimethylsiloxanes with molecular weights greater than the critical value for entanglement (M,). Data covering a frequency range from to lo8 Hz are reported for five samples and the frequency dependence of the modulus and shear viscosity in the terminal region are compared with the predictions of the shifted Rouse and Doi-Edwards models.Provided that allowance is made for the molecular-weight distribution of the polymer sample being studied it is found that for broad-molecular-weight-distribution samples both approaches are adequate. However for high-molecular-weight narrow-fraction samples it is found that the Doi-Edwards theory provides a better fit than the shifted Rouse model. Comparison of the data at high frequency with experiment indicates that the use of a Rouse model to describe the intermediate region of the viscoelastic spectrum is inappropriate. Suggestions are put forward as to the origins of the difference between experiment and theory.Viscoelastic measurements of a large number of polymers have indicated that for high-molecular-weight materials the low-frequency contribution to the relaxation spectrum is independent of the polymer type and is simply a function of its molecular weight.l? Studies of the variation of viscosity with molecular weight have further indicated that the behaviour of polymer melts can be subdivided into at least two regions polymers with molecular weight below Mc and those above. For the lower-molecular-weight polymers the viscosity is observed to vary approximately as the first power of the molecular weight whereas above Mc a 3.5power law is obeyed. A review by Graessley in 19743highlighted the poor theoretical understanding of the viscoelastic behaviour of polymers and subsequently prompted consideration of this problem by de Gennes4 and later by Edwards and DO^.^ In 1976 de Gennes4 proposed that the viscoelastic spectrum could be divided into two parts; a lower-frequency part associated with the diffusion-reptation of a polymer molecule through a matrix of polymer molecules and a higher-frequency contribution which corresponds to the motion of the chains between entanglement points formed as a consequence of polymer-polymer contacts.A rigorous analysis of this type of motion was developed t Present address Agricultural Research Council Food Research Institute Colney Lane Norwich NR4 7U4. 103 VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES by Doi and Edwards5 in 1979 and it describes the reptation-diffusion motion of a polymer with a molecular mass above M within an imaginary tube formed by the entanglement points.This motion models the so-called terminal relaxation3 region and constitutive equations describing viscoelastic response of such a system were deri~ed.~ These equations do not include contributions associated with relaxation of the chains between entanglement or the motion of local elements of the chain associated with the glass-liquid transition. These latter contributions to the overall motion have not been considered theoretically and semi-empirical relations are usually invoked to describe the higher-frequency contribution to the viscoelastic spectrum.3 In this paper an attempt is made to compare the results of isothermal viscoelastic experiments with the predictions of the reptation theory.Use of isothermal conditions rather than the superposition conditions usually used in viscoelastic studies1 avoids uncertainties with regards the effects of temperature on the form of the relaxation distribution. Data will be presented on a series of polymers with varying molecular weights and molecular-weight distributions. We believe this to be the first systematic attempt to test the ,validity of the reptation theory using viscoelastic measurements. EXPERIMENTAL MATERIALS A number of polydimethylsiloxane (PDMS) samples were used in this study. The first group of materials designated A had a broad molecular-mass distribution and were obtained from Midland Silicones Ltd.These samples were originally used by Dr G.Harrison for a study of the viscoelastic response of polydimethylsiloxanes at frequencies above lo4 HZ.~ The second group of samples designated B were obtained from Dr D. Meier of Midland Macromolecular Institute Michigan U.S.A. Fractions of broader-molecular-weight materials were obtained using gel-permeation chromatographic equipment at York University and are designated C. The apparatus has been described previ~usly,~ the columns used for the fractionation were obtained from Polymer Laboratories PLC. The column was eluted with toluene and the injection aliquot was typically 1-5 g ~m-~. The total elution volume was split into fractions and ca. 0.01-0.2 g of material were obtained in each fraction.Because of the severe practical limitations set by the small volumes which could be collected viscoelastic studies of the narrow fractions were limited to the low-frequency range. One of the samples designated D was prepared by ring-opening polymerisation of the cyclic trimer ;the conditions used are described CHARACTERISATIONOF THE POLYMERS The molecular masses of the polymers used in this study were determined using gel-permeation chromatography. The calibration of the columns used has been described previously.l0? l1This involved both osmotic pressure and light-scattering measurements of lower-molecular-mass samples and allowed an absolute calibration of the g.p.c. columns. A detailed discussion of the characterisation of linear and cyclic PDMS has been presented elsewhere.12 VISCOSITY MEASUREMENTS The viscosities of the samples were determined using a falling-ball method13 and are considered to be accurate to f5%.All measurements were made at 296.2 K. RESULTS The molecular masses molecular-mass distributions and viscosities are presented in table 1. The variation of viscosity with the number average molecular weight is shown in fig. 1 which also contains earlier results of the group at York UniversitylO. l2 R.R. RAHALKAR et al. Table 1. Molecular mass molecular-mass distributions and viscosities of linear pol ydimet hylsiloxanes code designate 10-3 M 10-3 M viscosity/Pa s s1 A 28.3 65.5 2.31 12.5 s2 A 44.9 84.4 1.88 34 s3 A 66.0 123.4 1.87 94 s4 B 84.7 150.2 1.88 387 s5 B 132.9 227.6 1.71 713 S6 D 93.5 108.2 1.15 1066 s7 C 92.6 124.0 1.34 1830 S8 C 84.6 114.7 1.36 47.6 s9 D 189.0 248.1 1.31 1430 3 2 1 h m CL .t v 0 go -1 -2 -3 1% M Fig. 1. Viscosity-molecular-mass relationship for linear polydimethylsiloxanes. Code as table 1. Undesignated points taken from ref. (1 1) and (1 2). VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES -1 0 2 4 6 8 log CfIW Fig. 2. Variation of modulus with frequency for sample S3. 0,G’(cu);0, G”(o);(-) Doi and Edwards theory and Rouse theory combined; (--) Rouse theory two-block model; (* * -* a) Doi and Edwards theory and Rouse theory combined monodisperse polymer; (-.-.) Doi and Edwards theory and modified Rouse theory combined.for the viscosity-molecular-mass relationship for PDMS of narrow molecular mass distribution and masses below M,. EXPERIMENTAL DYNAMIC VISCOELASTIC RESPONSE A torsional rheometer was used to make measurements of the components G’(o) and G”(co)of the shear modulus in the frequency range 10-1-102 Hz. The results for five of the polymers are shown in fig. 2-6. A description of the instrument has been published previo~s1y.l~ It consists essentially of an oscillatory cone-and-plate visco- meter. The errors in the values of G’(o) and G”(o)obtained are estimated to be < +5%. For certain of the samples the components R and XL of the shear impedance were determined at frequencies of 38 and 77 kHz by means of torsional quartz crystals; the method and apparatus used15+ l6have been described previously.Errors in the values of R and X are estimated to be < f5%. The components of the shear modulus were calculated using the equations G’(co)= (Ri-XE)/p (1) G”(co)= 2RLXL/p (2) where p is the density of the liquid. Where appropriate results obtained from the previous studys were combined with the data obtained in this study. Inclined incidence R. R. RAHALKAR et al. 107 7 c 5 4 h a" -. %? -M 1 1 0 -1 Fig. 3. Variation of modulus with frequency for sample S2. (-) Doi-Edwards and Rouse theory combined; (--) Rouse theory two-block model. 1 I I I I -1 0 2 lr 4 0 log CflW Fig. 4. Variation of modulus with frequency for sample S4.Symbols as fig.3. VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES 2 1 0 -1 Fig. 5. Variation of modulus with frequency for sample S7. Symbols as fig. 3. 't Fig. 6. Variation of modulus with frequency for sample S9. Symbols as fig. 3. R. R. RAHALKAR et al. I I I I I 0 2 4 6 8 log CfIHd Fig. 7. Variation of viscosity with frequency for sample S3. Symbols as fig. 2. measurements were performed on certain of the samples between 6 and 78 MHz together with R measurements using a normal incidence wave reflection technique at 450 MHz.” Within the limits of experimental error (ca. +6%) these results for the narrow- fraction materials are the same as those found previously,s confirming the observation that at this temperature (296.2 K) the viscoelastic behaviour of long-chain poly- dimethylsiloxanes is independent of chain length for frequencies > ca.lo7Hz. In making this comparison the method of reduced variables was used to allow for the small temperature difference between previous and present results i.e. values of R,/dp and XL/dpwere plotted as a function OfflV296.2 K/27303.2 K). To allow a later comparison with theoretical curves the variations of G’(o) and G”(w)have been calculated from the smoothed shear-impedance data. Previous resultss obtained for polymers having a broad molecular-mass distribution may not be applicable for the narrow-molecular-mass-distribution samples and are not included in the plots for the fractions.Only values of R are obtained from the measurements made at 450 MHz. In using eqn (1) to calculate G’(co)at this frequency it has been assumed on the basis of previous work that Xi < RL giving G’(o) xRL/p. Extrapolation of the results obtained at the lower frequencies indicates that XLat 450 MHz is probably ca. 0.4R.If this estimate is included in the calculation of G’(w),log G’(w) is reduced by only 0.08. Also for XL= (d2-1) R, G”(co)= G’(o); therefore at this frequency it is reasonable to assume that G”(co)xG’(cu). The variation of the dynamic viscosity with frequency ~’(cu) = G“(w)/2nf for the polymers studied is shown in fig. 7-1 1. As before the earlier measurementss for the broad-molecular-mass-distribution samples are included where appropriate.VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES I I I I I 0 2 4 6 8 log WHz) Fig. 8. Variation of viscosity with frequency for sample S2. Symbols as fig.3. 2 1 h LQ a -. 0 F W 00 --1 -2 I I 1 1 I 0 2 4 6 8 log CflW Fig. 9. Variation of viscosity with frequency for sample S4. Symbols as fig. 3. R. R. RAHALKAR et al. 1 1 I I I I 0 2 4 6 8 log CfIW Fig. 10. Variation of viscosity with frequency for sample S7. Symbols as fig. 3. -1 \ t 1-1 0 1 2 1 log CfIW Fig. 11. Variation of viscosity with frequency for sample S9. Symbols as fig. 3. DISCUSSION MOLECULAR-MASS DEPENDENCE OF THE VISCOSITY The molecular-mass dependence of the viscosity (fig. 1) can be subdivided into three regions.In region (i) low-molecular-mass polymers M < lo3 exhibit an M2 dependence. This behaviour is typical of non-polymeric liquids.8 There is evidence from previous studies’ that a broading of the viscoelastic relaxation occurs with increasing molecular mass. However at least ten repeat units are necessary in the chain for a Gaussian distribution of the end-to-end distance of segments. In region (ii) VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES 2 x lo3 <Zn<2.1 x lo4 the chain length is sufficient for the valid application of Gaussian statistics and the viscosity is observed to depend on the first power of the molecular mass. In region (iii) for polymers above 2 x lo4 the line is fitted to the data for polymers with values of Hw/Mn< 1.35.Broader-molecular-mass samples tend to lie to the left of this line. Taking this data selection we find that the viscosity varies as Mnz3.5. This dependence is in contrast with the Doi and Edwards theory which predicts an 3.0 beha~iour.~ The value of M from these data is ATn =21 000. This value is used in the subsequent discussion. DYNAMIC VISCOELASTIC BEHAVIOUR OF PDMS The total spectrum can be sub-divided into two regions the terminal region lying between and lo4 Hz and the intermediate region between lo4 and los Hz. The higher-frequency region associated with the glass-liquid transition is experimentally inaccessible for this polymer and lies above los Hz.’ This paper will concentrate on a discussion of the terminal and intermediate regions.A more complete presentation of the data and discussion of the theory used in modelling these experiments is to be published elsewhere.8 TERMINAL-REGION RELAXATION The Doi-Edwards theory5 predicts that the frequency dependence of the complex shear viscosity should be described by the following relationship 8 q*(iw) =-Go c -Td W (3) 5 PODD p4z21 i-iw G/p2 where Gtv is the average value of the equilibrium modulus wiis the mass fraction of molecules having molecular mass Mi, Gi is the diffusion time for the polymer within the tube and p is the mode number of the motion. This equation has been discussed fully by Doi and Edwardss and its application to viscoelasticity has been considered elsewhere.8 The above relationship does not allow for a molecular-mass distribution and as we can see from fig.2 the predictions of the theory based on the above relationship are unable to describe the observed experimental response. If we assume that where p =M/Mnthen eqn (3) can be transformed into m)is the mass fraction of molecules having molecular-mass ratios in the interval p-(p +Ap). Information on the distribution of p is directly obtained from g.p.c. data and evaluation of eqn (5) may be performed by summation of the mode contributions for each of the molecular-mass components present. In the computation we recognise that G cc M3in which case G in eqn (5) can be written in the form T,p3.Using the relationship that G*(iw) =G’(w)+iG”(w) =iwq*(iw) (6) we can obtain and =coq’(w).R. R. RAHALKAR et al. The above equations describe the terminal region. To the values predicted by eqn (7) and (8) we require to add a contribution due to the motion of the chains between entanglement and this is usually described by a Rouse mode with molecular mass M,. A detailed discussion of this calculation together with the equations for the shifted Rouse mode are presented elsewhere.8 It is important to realise that the simple Doi and Edwards model does not describe the complete viscoelastic response of a polymer and that terms associated with motions between entanglements and local motions of the polymer backbone have to be added to the terminal-region contribution in order for the whole spectrum to be described correctly. The contribution from these higher-frequency modes is very small in the frequency range 10-1-103 Hz; however they dominate the response observed at higher frequencies.The existence of these higher-frequency modes must not be forgotten when attempting to model the time-dependent behaviour of a polymer melt whatever the technique being used. It can be seen from fig. 2-4 that for these broad-molecular-mass-distribution samples the fit of the data is good using either the shifted Rouse or the Doi and Edwards model. Similarly the narrow-molecular-mass fraction S7 (fig. 5)can be fitted by either theory adequately. It is only with the very-high-molecular-weight sample S9 (fig. 6) that it is possible to differentiate between the two approaches. It is clear from fig.7-1 1 that the terminal-region relaxation is more accurately described by the Doi and Edwards theory. Unfortunately the experimental frequency range available without resort to time-temperature superposition is limited and it has not been possible to cover the complete relaxation region in this study. Efforts are currently being made to extend these measurements in an attempt to explore whether or not the theory can describe the form of the curve in the tail of the terminal region. In summary these experiments indicate that the Doi and Edwards theory modified as necessary to allow for the molecular-mass distribution is capable of describing the relaxation of polymers covering a molecular-mass range from M,% 21OOO (= M,) to M % 190000. INTERMEDIATE-RELAXATION REGION Relaxations above ca.lo4 Hz can be associated with the motion of the chain trapped between entanglements. Usually the frequency dependence of the viscoelastic relaxation in this region has been described by a Rouse type of equation with characteristic molecular mass equal to Me.It is clear from fig. 2 and 7 that this type of approach is unable to predict correctly the magnitude of the viscoelastic response in the frequency range 106-107 Hz. Attempts to obtain better agreement by using a lower molecular mass for the effective value of M used in the Rouse contribution lead to the prediction of too low an amplitude in the frequency range 104-105 Hz. In an attempt to explain the origins of this discrepancy we have developed a model in which the polymer chain is allowed to adopt an asymmetric distribution about the entanglement points rather than the symmetric one assumed by the simple Rouse approach (fig.12). The analysis of this situation will be presented elsewhere.* It is clear from fig. 2 and 7 that allowing an asymmetric distribution does increase the viscoelastic contribution in the megahertz frequency range with only a minor reduction in amplitude for the contribution in the kilohertz region. However the change in the distribution is greater than is required experimentally. The asymmetric distribution reduces the length of certain of the polymer tails which extend beyond an entanglement and increasing others. The data presented here would indicate that the distribution used overestimates the effects of these short chains and underestimates the number of the longer tails.A more complete discussion of this topic will be presented elsewhere.8 VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES (b) Fig. 12. Schematic representation of polymer motions. The dotted lines indicate interacting polymer chains. (a) Symmetrical distribution (b)assymetrical distribution. CONCLUSIONS This present study of polydimethylsiloxanes indicates that the Doi and Edwards model can describe the viscoelastic behaviour of polymer with molecular weights above M,. It must however be stressed that accurate modelling of experimental data is only possible when the effects of molecular-mass distribution and the presence of additional higher-frequency contributions are included correctly.It is also clear that the factors controlling the higher-frequency relaxation processes are not completely understood and it is probable that a more precise modelling of these motions may help to resolve the long-standing problem of the discrepancy between theory and experiment regarding the power law for the viscosity. R. R. R. and W. H. thank the S.E.R.C. for support during the period of this study. Provision of financial support allowing the purchase of certain items of electronics and also the columns used in the sample fractionation is also gratefully acknowledged. 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